80 THEORY OF THE MICROSCOPE. 



incident cone of rays is large enough to fill the whole aperture of 

 the objective. This condition is, however, very seldom fulfilled in 

 microscopic observations, since the diaphragm limits the inclination 

 of the incident rays. In place of co, therefore, 

 the angle a p b (Fig. 40) enters, at which the 

 diaphragm a b is seen from the point p in the 

 plane of adjustment. If the angle = 8, the 



/2S\ 2 

 --- above expression becomes v = lJ . With 



the same illumination, the brightness of the 

 field of view is, accordingly, in inverse propor- 

 tion to the squares of the coefficients of linear 

 amplification. If, for instance, we select a 

 diaphragm, so that B = 30, the coefficients of 

 amplification will be 240, 300, 360, 420, &c., for 

 the more powerful objectives, and the relative 

 brightness T V, -3-5, inr* iV> & c - But if, on the 

 other hand, we increase the value of S to double 

 or treble the assumed value (by using a larger 

 diaphragm), the brightness attains provided 

 always that o> is at least of the same magnitude 



four times and nine times, respectively, the above fractions ; 



these fractions will have, therefore, equal denominators, and the 



numerators 4 and 9. 



/2ft>\2 . 



For larger angles of aperture the expression v = ( ---) is not 



* 1f)\j ' 



quite correct. The quantity of light in the different cones of rays 

 is, strictly speaking, not in proportion to the squares of the 

 angular magnitude indicated, but to the area cut off' by the cone 

 from a sphere having its centre at the apex of the cone, that is, 

 to the area of the active calotte. For it is evident that a self- 

 luminous radiant emits rays in all directions, and therefore 

 illuminates equally at all points a spherical surface, whose centre 

 lies in the source of light itself. It depends, therefore, upon the 

 extent of this spherical surface which is effective in a given case ; 

 an ?i-times as great an area of this calotte always corresponds to 

 an 7i-fold amount of light. If we work out the calculation for 

 different apertures of the incident cones of rays, and take as unity 

 the amount of light for co = J, we obtain, for example, the values 

 given in the following table. The fourth column contains the 



